TY - JOUR

T1 - Transience, recurrence and critical behavior for long-range percolation

AU - Berger, Noam

PY - 2002/4

Y1 - 2002/4

N2 - We study the behavior of the random walk on the infinite cluster of independent long-range percolation in dimensions d = 1, 2, where x and y are connected with probability ∼ β/∥x - y∥-s. We show that if d < s < 2d, then the walk is transient, and if s ≥ 2d, then the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension d ≥ 1, if d < s < 2d, then there is no infinite cluster at criticality. This result is extended to the free random cluster model. A second corollary is that when d ≥ 2 and d < s < 2d we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network.

AB - We study the behavior of the random walk on the infinite cluster of independent long-range percolation in dimensions d = 1, 2, where x and y are connected with probability ∼ β/∥x - y∥-s. We show that if d < s < 2d, then the walk is transient, and if s ≥ 2d, then the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension d ≥ 1, if d < s < 2d, then there is no infinite cluster at criticality. This result is extended to the free random cluster model. A second corollary is that when d ≥ 2 and d < s < 2d we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network.

UR - http://www.scopus.com/inward/record.url?scp=0036011687&partnerID=8YFLogxK

U2 - 10.1007/s002200200617

DO - 10.1007/s002200200617

M3 - Article

AN - SCOPUS:0036011687

SN - 0010-3616

VL - 226

SP - 531

EP - 558

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -